Philosophy 167: Class 14 - Part 7 - Newton-Flamsteed Correspondence: Flamsteed on the Puzzlingly Regular Motions of the Jovian Moons, and Newton on the Possible Interaction Between Jupiter and Saturn.

Synopsis: Letters between Flamsteed and Newton show that the orbits of Jupiters moons follow the three halves power law of Kepler, if you account for the speed of light as measured by Romer. They also speculate on gravitational interaction between Jupiter and Saturn.

All right, what happens is, Newton starts a correspondence sometime in December of 1684 with Flamsteed. We do not have the first letter between the two of them. Newton sends request to Flamsteed, who is the Royal Astronomer, and Flamsteed answers with the answers. So just on this page, I'll only look a moment.

It's the next page I'll start focusing. Mr. Halley delivered me a note wherein you desire me to give you the places of two fixed stars in the foot of Perseus. And Mr. Paget, I don't know how to pronounce his name, soon after a letter from you, wherein you kindly offered me the perusal of your papers, for which I humbly thank you.

Mr. Paget promised to be here on Tuesday last, but the severity of that day's weather I suppose prevented him. Else I had sent you by him the places of the stars, which I now transmit. Okay, why would Newton be asking for places in the stars? He needs them for comets.

That was one of his problems with the comet of 1680, 81. He could do observations himself, other people did observations. They related him to the locations of specific stars. But where are those stars in the bigger picture? That's what he's asking for. Then it turns out there's another question.

In your letter to Mr. Paget, you further require my determinations of the distances of Jupiter satellites from him and their periods. I gave their utmost elongations in numbers 96 of Mr. Oldenburg's transactions. That's what we call a fill transaction, he's still calling them Mr. Oldenburg's. Oldenburg had died in 1677 so showing respect.

But having since that time made many observations of their eclipses, I find they require them something larger and they are as exactly in the sequi, he misspelled it. Sesquialtera proportion to their periods as it is possible for our senses to determine. So this is what Newton wanted to know, satellites of Jupiter, three halves power rule will hold, and if so, how exactly?

And there is worth including in here the note down below as he gives you the things, he gives you the data, etc. Revolutions and then says, I use their motions altogether equable, only allowing Romer's equation of light, without which allowance the errors of my tables would be above ten minutes of time.

Now it seems strange the moon's motions should be so perplexed with inequalities and these for aught I can perceive yet except in the second, wholly free from them. And I have some reason to think the errors I meet with in my numbers for the second satellite, may partly proceed from my having allowed its orbit to lie in the same plane with the orbits of the other three.

Whereas I have some observations that will scarce allow it to lie otherwise but near him. Okay. Do notice what this is being said. First of all it's being said you gotta include the equation for Romer's equation for light in order to have the satellites of Jupiter to be well-behaved.

If you do, three halves power rule holds perfectly. But then he asked a very good question, I don't know if Newton had already thought of this question. If he had, either way, it's gonna become very important to Newton. It's gonna become one of the central questions of the Principia.

But whether he thought of it, or Flamsteed is here provoking him. Here's the puzzle, our moon's motion was well known, was too complicated for anybody to ever to have been able to calculate. It has inequalities on top of inequalities, most of which, or at least some of which had never been characterized, okay?

They only had principal inequalities and they knew there was further complications. No sign of that whatsoever in the motions of Jupiter's satellites. Why should those satellites be so perfect, and our moon's be so bad? Okay. That's a good question. And then, this last thing, just to throw out, Newton had also asked about the satellites of Saturn.

And I'm not going to read this one, he simply tells you, I haven't been able to observe any but Huygen's satellite Titan. The three others that Cassini has announced, Flamsteed has been unable to detect with his equipment. That's a comment about how much better the Campari telescopes actually were, the telescopes in England.

That is, Cassini simply couldn't pick them out with his telescopes. So, what happens in the first edition of the Principia, Newton treats Saturn as having only one satellite. By the time of the second edition, he changes that, because by then, they've managed to confirm, other than Cassini observing the multiple satellites, he said.

Of course, they knew Saturn had a ring. That's different. All right, Newton replies to this. That was 27th January. Newton's reply is 30, December. And I repeat Flamsteed's in Greenwich which is a half hour boat ride, 40 minute boat ride from London. And Newton is in Cambridge which is an hour and five minute train ride from London.

But three days later, it's back in the mail, so the mail was pretty good at the time. And what does Newton say, I underlined all the parts that I'm concerned with. I put this whole correspondence on supplementary material, the whole thing. The orbit of Saturn is defined by Kepler too little for the sesquialtera proportion.

In other words, he's looking at Kepler, and concluding Saturn doesn't fit the three halves power rule. Now we can understand why Saturn's a problem, right, 29 year period. Tycho didn't even have data for 29 years. The period didn't extend that long, till he got moved from Hven into middle Europe, into Prague.

And he didn't do that much observation in Prague. One claim is because he was too busy drinking all the time at parties but regardless, there was not much observation. So there's a whole question here of what Saturn's orbit should be. It's not clear at this point. Continuing, this planet so often seen is in conjunction with Jupiter, ought by reason of Jupiter's action upon him to run beyond his orbit about one or two of the sun's semi diameters.

Or a little more, almost all the rest, excuse me. One or two semi diameters or a little more. And almost all the rest of his motion to run as much or more within it. Perhaps that might be the ground of Kepler's defining it too little, but I would gladly know if you ever observe Saturn to err considerably from Kepler's tables about the time of his conjunction with Jupiter.

So let's back up a moment. He had just concluded in the Copernican Scholium, the motions are too complicated to calculate. What's he doing now, he's trying to calculate the perturbation of the motions. Very typical of new Newton, they may be too complicated to calculate but that doesn't just stop him from trying to figure it out.

Yeah, he's equally asking, does the Jupiter tendency reach all the way to Saturn? Saturn is almost as far away from Jupiter as the sun is from Jupiter. Saturn's orbit is five, essentially five astronomical units. Saturn's is nine something. I actually brought, Chip tends to know these things, but I brought my standard handbook for once tonight to be able to get the correct number on Saturn.

There's a nice table in here on all the orbital motion in our planetary system. So, it's a simple several-paged one to look at. So, radius of the orbit is 9.6, 9.57 astronomical units. So 5.2 to 9.5 is 4.3 astronomical units and Jupiter is 5.2. So if Jupiter's influence reaches Saturn, surely it reaches the Sun too.

So what he may be doing here is simply trying to confirm that these things are acting on the Sun by showing that they act on Saturn, okay? That's gonna become a very, very celebrated problem, the action of Jupiter on Saturn, and Saturn on Jupiter. It gets resolved in 1785 by Laplace, and it is the last shoe falling In the confirmation of Newtonian gravity, okay?

It's the very last nail in the coffin for it, it's large. The effect on Saturn is almost one degree displacement. What made it so hard to detect and nobody could figure this out till Laplace did, the period of the one degree effect is 900 years. If you're looking at it for observation, you're gonna need at least 500 years of observation to see what the pattern is.

So it's nice Laplace figured it out without having to go through the observations. He was able to do the calculation Newton can't really do, and you'll see why later in the course. Down to the last remark in here, but I would gladly know, yeah, I already did that.

For I would glad know the proportion of the orbits of the satellites to that of Jupiter as exactly as I can. This is fairly clear what he wants, by my account. He wants his precise value of this centripetal tendency towards Jupiter as he can get, because he has one that's very precise toward the sun.

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